Question: The geometric sequence $a_i$ is defined by the formula: $a_1 = 8$ $a_i = a_{i - 1} \cdot \dfrac34$ Find the sum of the first $25$ terms in the sequence. Choose 1 answer: Choose 1 answer: (Choice A) A $0.03$ (Choice B) B $4.57$ (Choice C) C $ 31.98 $ (Choice D) D $4.05\cdot10^{21}$
Answer: Getting started Let's write out the first few terms of the series: $8 + 6 + 4.5...$ We're dealing with a geometric series because each term is multiplied by $\dfrac34$ to get the next term. We need a formula to compute the sum of the terms. Formula for geometric series The sum $S_n$ of a finite geometric series is $S_n = \dfrac{a_1(1-r^n)}{1-r}$ where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. What do we need to use the formula? The first term $(a_1 = {8})$ and the number of terms $(n = {25})$ are given in the question. The common ratio $r$ is ${\dfrac34}$ because each term is multiplied by ${\dfrac34}$ to get the next term. Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac{a_1(1-r^n)}{1-r} \\\\ S_{{25}}&=\dfrac{{8}\left(1-\left({\dfrac34}\right)^{{25}}\right)}{1-\left({\dfrac34}\right)} \\\\ S_{{25}}&=32\left(1-\left({\dfrac34}\right)^{{25}}\right) \\\\ S_{{{25}}} &\approx 31.98 \end{aligned}$ The answer $ 31.98 $